Hazard was dismissed in the closing minutes for a moment of madness in a 0-0 semi-final second-leg draw at the Liberty Stadium after being involved in a tussle for the ball with a ballboy.
With the ballboy delaying the game by smothering the ball when out of play, the Belgian stupidly kicked out at a combination of the ball and the boy's ribs.
It was the lowest point on a night to forget for Chelsea, as Swansea reached their first major cup final after a 2-0 first-leg win and they will now face League Two Bradford City at Wembley.
Swansea first had to survive a penalty scare after Demba Ba, starting ahead of Fernando Torres, went down under a challenge from Ben Davies.
Referee Chris Foy opted not to point to the spot although there was contact as Ba looked to step inside the full-back.
Michu, fresh from signing a new four-year contract, then tested Chelsea but found Petr Cech equal to his angled drive.
Angel Rangel cleared Gary Cahill's looping header off the line as the Chelsea pressure increased, while Ba wasted another opportunity late in the half as he scooped a shot over the bar.
The former Newcastle striker was again off-target with a curling strike from the edge of the area five minutes into the second half, but Michael Laudrup was by now imploring his defenders to get out of their own 18-yard box.
Frank Lampard and Juan Mata linked beautifully on the right, moments after taking part in an awful free-kick routine, but yet again there was a white shirt in the way to block the Spaniard's attempted ball across the six-yard box.
Gerhard Tremmel made a sharp stop to deny a fizzing effort from Hazard as the game entered the final 20 minutes, and Chelsea's hopes were effectively ended when Hazard was then dismissed in unsavoury fashion.
Pablo Hernandez and Nathan Dyer missed chances to secure victory on the night as Cech saved on both occasions, with the stadium still stunned by Hazard's astonishing dismissal.
And Swansea comfortably saw out six minutes of stoppage time to reach Wembley.